1

When you see…

Find the zeros

You think…

2

To find the zeros...To find the zeros...

Set function = 0 Factor or Graph to find zeros on calculator.

3

When you see…

Find equation of the line tangent to f(x) at (a, b)

You think…

4

Equation of the tangent lineEquation of the tangent line

Point and Slope (a, f(a)) m = f ’(a)

Use y- y1 = m( x – x1 )

5

You think…

When you see…

Find equation of the line normal to f(x) at (a, b)

6

Equation of the normal lineEquation of the normal line

Point and Slope (a, f(a)) m = - 1 f ’(a) Use y- y1 = m( x – x1 )

7

You think…

When you see…

Find the interval where f(x) is increasing

8

ff(x) increasing(x) increasing

Determine where f ’ (x) > 0

9

You think…

When you see…

Find the interval where the slope of f (x) is increasing

10

Slope of Slope of f f (x) is increasing(x) is increasing

Find where f ‘ is increasing…

Find where f “ (x) is positive

11

You think…

When you see…

Find the minimum value of a function

12

Local Minimum value of a functionLocal Minimum value of a function

Make a sign chart of f ‘( x)

Find x where f ‘( x) changes from - to + Plug those x values into f (x) Choose the smallest

13

You think…

When you see…

Find critical numbers

14

Find critical numbersFind critical numbers

Find f ‘ (x ) = 0 and f ‘ (x ) = undef.

15

You think…

When you see…

Find inflection points

16

Find inflection pointsFind inflection points

Find f ‘‘ (x ) = 0 and f ‘‘ (x ) = undef. Make a sign chart of f “ (x) Find where f ‘‘ (x ) changes sign ( + to - ) or ( - to + )

17

You think…

When you see…

Show that exists lim ( )x a

f x

18

Show existsShow exists lim ( )x a

f x

Show that

limx a

f x limx a

f x

19

You think…

When you see…

Show that f(x) is continuous

20

..f(x) is continuousf(x) is continuous

S h o w t h a t

1 ) xfax

lim e x i s t s ( p r e v i o u s s l i d e )

2 ) af e x i s t s

3 ) afxfax

lim

21

You think…

When you see…

Show that f(x) is differentiable at x = a

22

f(x) is differentiablef(x) is differentiable

Show that

1) f(a) is continuous (previous slide) 2) xfxf

axax

limlim

23

You think…

When you see…

Find vertical

asymptotes of f(x)

24

Find holes/vertical asymptotes of f(x)Find holes/vertical asymptotes of f(x)

Factor/cancel f(x)

VA: Set denominator = 0

Hole: where factor that cancels = 0

25

You think…

When you see…

Find horizontal asymptotes of f(x)

26

Find horizontal asymptotes of f(x)Find horizontal asymptotes of f(x)

Show

xfx lim

and

xf

x lim

27

You think…

When you see…

Find the average rate of change of f(x) at [a, b]

28

Average rate of change of f(x)Average rate of change of f(x)

Find

ARC = f (b) - f ( a)

b - a

29

You think…

When you see…

Find the instantaneous rate of change of f(x)

at x = a

30

Instantaneous rate of change of f(x)Instantaneous rate of change of f(x)

Find f ‘ ( a)

31

You think…

When you see…

Find the average valueof xf on ba,

32

Average value of the functionAverage value of the function

Find b

adxxf

ab)(

1

33

You think…

When you see…

Find the absolute maximum of f(x) on [a, b]

34

Find the absolute maximum/Min of f(x)Find the absolute maximum/Min of f(x)

Candidate Test 1. Test CP 2. Test Endpoints

35

You think…

When you see…

Show that a piecewise function is differentiable at the point a where the function rule splits

36

Show a piecewise function is Show a piecewise function is

differentiable at x=adifferentiable at x=a

F i r s t , b e s u r e t h a t t h e f u n c t i o n i s c o n t i n u o u s a t

ax .

T a k e t h e d e r i v a t i v e o f e a c h p i e c e a n d s h o w t h a t

xfxfaxax

limlim

37

You think…

When you see…

Given s(t) (position function),

find v(t)

38

Given position s(t), find v(t)Given position s(t), find v(t)

Find tvts

39

You think…

When you see…

Total Distance

40

Given v(t), find how far a particle Given v(t), find how far a particle travels on [a,b]travels on [a,b]

TD= b

a

dttv

41

You think…

When you see…

Find the average velocity of a particle

on [a, b]

42

Find the average rate of change on Find the average rate of change on [a,b][a,b]

Find

abasbs

ab

dttvb

a

43

You think…

When you see…

Given v(t), determine if a particle is speeding up at

t = k

44

Given v(t), determine if the particle is Given v(t), determine if the particle is speeding up at t=kspeeding up at t=k

Find v(k) and a(k). If v(k) and a(k) signs are the same, the particle is speeding up. If v(k) and a(k) signs are different, the particle is slowing down.

45

You think…

When you see…

Given v(t) and s(0),

find s(t)

46

Given v(t) and s(0), find s(t)Given v(t) and s(0), find s(t)

dttvsts 0

47

You think…

When you see…

Mean Value Theorem

48

Show that the MVT holds on [a,b]Show that the MVT holds on [a,b]

f is continuous and differentiable on the interval.

Then the slope of tangent line = slope of secant line

f cf b f a

b a.

49

You think…

When you see…

Find f ’(x) by definition

50

Find f Find f ‘‘( x) by definition( x) by definition

f x limh0

f x h f x h

or

f x limx a

f x f a x a

51

You think…

When you see…

Find the derivative of the inverse of f(x) at x = a

52

Derivative of the inverse of f(x) Derivative of the inverse of f(x)

))(('

1)(

11

xffxf

dx

d

.

53

You think…

When you see…

y is increasing proportionally to y

54

.y is increasing proportionally to yy is increasing proportionally to y

kydt

dy

translating to

ktCey

55

You think…

When you see…

The rate of change of population is …

56

Rate of change of a population Rate of change of a population

...dt

dP

57

You think…

When you see…

dttfdx

d x

a

58

Fundamental TheoremFundamental Theorem

2nd FTC: Answer is xf

59

You think…

When you see…

)(

)(xu

adttf

dx

d

60

Fundamental Theorem, againFundamental Theorem, again

)()(' xufxu

61

You think…

When you see…

Integrate

62

1. Estimation:LRAMRRAM (Riemann Sums)MRAMTrapezoid

2. Geometry

3. Antiderivative FTC

4. Calculator

Methods for IntegrationMethods for Integration

63

You think…

When you see…

Find area using Riemann/Trapezoidal

sums

64

Area using Riemann sumsArea using Riemann sums

)()( rlrR xfxxA

)()( llrL xfxxA

)()( mlrM xfxxA

)()()(2

1lrlrT xfxfxxA

65

You think…

When you see…

Solve the differential equation …

66

Solve the differential equation...Solve the differential equation...

Separate the variables – x on one side, y on the other. Separate only by multiplication or division and the dx and dy must all be upstairs..

67

You think…

When you see…

Meaning of

dttfx

a

68

Meaning of the integral of f(t) from a to xMeaning of the integral of f(t) from a to x

The Area accumulation function – accumulated area under the function xf starting at some constant a and ending at x

69

You think…

When you see…

Given a base, cross sections perpendicular to

the x-axis that are

70

The Slicing Method

dxxAVb

a )(

71

You think…

When you see…

Find where the tangent line to f(x) is horizontal

72

Horizontal tangent lineHorizontal tangent line

Set xf = 0

73

You think…

When you see…

Find where the tangent line to f(x) is vertical

74

Vertical tangent line to f(x)Vertical tangent line to f(x)

Find xf . Set the denominator equal to zero.

75

You think…

When you see…

Find the minimum

acceleration given v(t)

76

Given v(t), find minimum accelerationGiven v(t), find minimum acceleration

First find the acceleration tatv Minimize the acceleration by

Setting ta = 0 or ta = undef. Then determine where ta changes

from – to +

77

You think…

When you see…

Approximate the value f(c) of by using the

tangent line to f at x = a

78

Approximate f(c) using tangent line Approximate f(c) using tangent line to f(x) at x = 0to f(x) at x = 0

Plug c in for x in tangent line

79

You think…

When you see…

Find the exact value f(c)

80

Plug c in for x in original equation

81

You think…

When you see…

Given the value of F(a) and the fact that the

anti-derivative of f is F, find F(b)

82

FTCFTC

aFbFdxxfb

a

.

83

You think…

When you see…

Find the derivative off(g(x))

84

Find the derivative of f(g(x))Find the derivative of f(g(x))

xgxgf

Think . . . Chain Rule

85

You think…

When you see…

Given a graph of

find where f(x) is

increasing

'( )f x

86

Given a graph of f ‘(x) , find where f(x) is Given a graph of f ‘(x) , find where f(x) is increasingincreasing

Determine where xf is positive

87

You think…

When you see…

Given a chart of x and f(x) on selected values between a and b, estimate where c is between a and b.

'( )f x

88

MVT

ab

afbfcf

89

You think…

When you see…

Given , draw a

slope field dx

dy

90

Draw a slope field of dy/dxDraw a slope field of dy/dx

Use the given points

Plug them into dx

dy,

drawing little lines with theindicated slopes at the points.

91

You think…

When you see…

Find the area between curves f(x) and g(x)

92

Area between f(x) and g(x) on [a,b]Area between f(x) and g(x) on [a,b]

dxbottomtopAr

L

x

x

dyleftrightAt

b

y

y

93

You think…

When you see…

Volume of Revolution

94

dxxrxRVb

a

washer 22 )()(

dxxrVb

a

disk 2)(

(1) dx or dy, (2) washer or disk, …